The Phugoid model is a system of two nonlinear differential equations
in a frame of reference relative to the plane. Let v(t) be the speed
the plane is moving forward at time t, and (t) be the angle
the nose makes with the horizontal. As is common, we will suppress
the functional notation and just write v when we mean v(t), but it
is important to remember that v and are functions of time.

If we apply Newton's second law of motion (force =
mass × acceleration) and examine the major forces acting on the
plane, we see easily the force acting in the forward direction of the
plane is

m = - mg sin - drag.

This matches with our intuition: When is negative, the nose
is pointing down and the plane will accelerate due to gravity. When
> 0, the plane must fight against gravity.

In the normal direction, we have centripetal force, which is often
expressed as mv2/r, where r is the instantaneous radius of
curvature. After noticing that that
= v/r,
this can be expressed as
v, giving

mv = - mg cos + lift.

Experiments show that both drag and lift are proportional to v2,
and we can choose our units to absorb most of the constants. Thus, the
equations simplify to the system

= - sin - Rv2 =

which is what we will use henceforth. Note that we must always have v > 0.

It is also common to use the notation for
and
for
. We will use these notations
interchangeably.